zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Efficient iterative solution of the three-dimensional Helmholtz equation. (English) Zbl 0929.65089
The authors examine two types of preconditioners for the discrete indefinite Helmholtz equation with Sommerfeld-like boundary conditions. The first is derived by discretization of a related continuous operator, the second uses the block Toeplitz approximation to the desired problem. The resulting preconditioning matrices allow the use of fast transform methods (e.g. fast Fourier transform) and differ from the discrete Helmholtz operator by an operator of low rank. Some numerical experiments presented in the paper demonstrate the efficiency of the method when combined with Krylov subspace iteration. The authors show that the performance of restared GMRES with the proposed preconditioners is relatevely insensitive to the discretization mesh size and the wave number, and the algorithms are highly parallelizable. The presented technique is potentially applicable to inhomogeneous media, exterior domain problems and non-Cartesian grids.
65N06Finite difference methods (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
65F10Iterative methods for linear systems
65F35Matrix norms, conditioning, scaling (numerical linear algebra)